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Friday, January 5, 2007

Making Platonic Solids


Today I am sharing with you a very interesting creative mathematics project on Platonic solids. Actually, yesterday one of my colleagues son came to the school for doing a project on mathematics . He made beautiful 5 platonic solids using transparent sheets. I explained to him the procedure of making them.In the last period he came to me to show his work.I was amazed to see the result. There, I thought of clicking the photographs on my mobile and I requested him to show each and every step one by one .He said I am camera shy! But later agreed upon.


The first question which comes into the mind is What are platonic solids?

A platonic solid is a polyhedron all of whose faces are congruent regular polygons and where the same number of faces meet at every vertex.There are 5 platonic solids as shown below.


They are a cube, a tetrahedron, an octahedron, a dodecahedron and an icosahedron.
The Greeks recognized that there are only five platonic solids. But why is this so? The key observation is that the interior angles of the polygons meeting at a vertex of a polyhedron add to less than 360 degrees. To see this note that if such polygons met in a plane, the interior angles of all the polygons meeting at a vertex would add to exactly 360 degrees. Now cut an angle out of paper, and fold another piece of paper to that angle along a line. The first piece will fit into the second piece when it is perpendicular to the fold. Think of the fold as a line coming out of our polyhedron. The faces of the polyhedron meet at the fold at angles less than 90 degrees. How can this be possible? Try wiggling your first piece of paper within the second. To be able to incline it with respect to the fold you have to decrease the angle of the first piece, or increase the angle of the second.
Triangles. The interior angle of an equilateral triangle is 60 degrees. Thus on a regular polyhedron, only 3, 4, or 5 triangles can meet a vertex. If there were more than 6 their angles would add up to at least 360 degrees which they can't. Consider the possibilities:
3 triangles meet at each vertex. This gives rise to a Tetrahedron as shown below.
4 triangles meet at each vertex. This gives rise to an Octahedron as shown below.
5 triangles meet at each vertex. This gives rise to an Icosahedron as shown below.
Since the interior angle of a square is 90 degrees, at most three squares can meet at a vertex. This is indeed possible and it gives rise to a hexahedron or cube as shown below.

Pentagons. As in the case of cubes, the only possibility is that three pentagons meet at a vertex. This gives rise to a Dodecahedron as shown below.

Hexagons or regular polygons with more than six sides cannot form the faces of a regular polyhedron since their interior angles are at least 120 degrees.
That was mathematics. Now CREATIVE MATHEMATICS......
Now I will explain you the procedure of making them.
For making the platonic solids we have used transparent sheets(OHP sheets), a pair of scissors, a marker , cello tape , compass and a ruler.
Let us make a tetrahedron.
Step 1 Take the net of the tetrahedron (You can download the nets of solids from any website on Platonic solids).
Step 2 Place an OHP sheet(transparency) on it.

Step 3 Using a compass and a ruler scratch the outline of the net.

You will get the impression shown below.

Step 4 Cut the outer boundary of the impression on the OHP sheet.You will get something like this. (Shown below)
Step 5 Fold along the scratched lines.
Use a cellotape and cover the outer flaps.


You can make the solids colourful and attractive by filling it with some dry flowers or coloured balls etc.

Put a cellotape on the last flap to get the tetrahedron.

Thus you have seen that, how interesting it is to do it.
After making the solids , experiments on these solids can be done. Various properties of each of them can be further explored by the students themselves.I think ,it will definitely create interest in them to learn more about them due to their aesthetic power.
One day I was reading a spiritual magazine in which I saw a picture of platonic solids.I got very much surprised. When I explored the web pages , I found that they are used for meditation and other enchanting purposes. They are considered as a good 3-Dimensional symbols.
Interesting na!
Bye!

6 comments:

Anonymous said...

BEAUTIFUL PLATONIC SOLIDS.NICELY EXPLAINED.YOU ARE A GREAT TEACHER.

Daniel said...

Hey, if anyone wants more information about the Platonic solids, check out my site:

www.PlatonicSolids.info

It contains lots of information, and cool downloadable graphics and animated GIFs.

Dan Radin

TOM said...

u r a very good teacher u have helped a lot my explaining platonic solids.thanks

vivek singh said...

thank you i needed it

Kunal Agarwal said...

it helped a lot.........thank you very much

shalvi said...

yeah, very helpful for school project,

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